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Let $V_{\lambda}$ and $V_\mu$ be representations of the symmetric groups $\mathfrak{S}_d$ and $\mathfrak{S}_m$ respectively where $\lambda$ is a partition of $d$ and $\mu$ is a partition of $m$. It is claimed in Fulton and Harris that the induced representation of the tensor product of these two representations (which is a representation on $\mathfrak{S}_{d+m}$) satisfies the decomposition $V_\lambda \circ V_\mu = \sum{N_{\lambda\mu\nu}V_\nu}$, where $N_{\lambda\mu\nu}$ are the Littlewood-Richardson coefficients and the sum is over all partitions $\nu$ of $d+m$.

In the preceeding exercise, one can show that the irreducible character $\chi_\lambda$ can be written as a determinant of induced characters, that is $\chi_\lambda = \left|\psi_{\lambda_i + j - i} \right|$. This follows in large from Giambelli's formula.

The text claims that the the decomposition of the induced character above can be derived by simply multiplying the determinantal formula. I don't see why this is true - the tensor of the two representations certainly has a character that is a product of the two, but why can we obtain the character of the induced representation of a tensor of representations by multiplying the characters of the representations being inducted?

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